WORST_CASE(?,O(n^1)) * Step 1: Sum WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: goal(x,xs) -> member(x,xs) member(x,Nil()) -> False() member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x',x),x',Cons(x,xs)) notEmpty(Cons(x,xs)) -> True() notEmpty(Nil()) -> False() - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs) member[Ite][True][Ite](True(),x,xs) -> True() - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ,goal,member,member[Ite][True][Ite] ,notEmpty} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: goal(x,xs) -> member(x,xs) member(x,Nil()) -> False() member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x',x),x',Cons(x,xs)) notEmpty(Cons(x,xs)) -> True() notEmpty(Nil()) -> False() - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs) member[Ite][True][Ite](True(),x,xs) -> True() - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ,goal,member,member[Ite][True][Ite] ,notEmpty} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): The following argument positions are considered usable: uargs(member[Ite][True][Ite]) = {1} Following symbols are considered usable: {!EQ,goal,member,member[Ite][True][Ite],notEmpty} TcT has computed the following interpretation: p(!EQ) = 0 p(0) = 0 p(Cons) = 0 p(False) = 0 p(Nil) = 0 p(S) = 0 p(True) = 0 p(goal) = 4 p(member) = 0 p(member[Ite][True][Ite]) = x1 p(notEmpty) = 0 Following rules are strictly oriented: goal(x,xs) = 4 > 0 = member(x,xs) Following rules are (at-least) weakly oriented: !EQ(0(),0()) = 0 >= 0 = True() !EQ(0(),S(y)) = 0 >= 0 = False() !EQ(S(x),0()) = 0 >= 0 = False() !EQ(S(x),S(y)) = 0 >= 0 = !EQ(x,y) member(x,Nil()) = 0 >= 0 = False() member(x',Cons(x,xs)) = 0 >= 0 = member[Ite][True][Ite](!EQ(x',x),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) = 0 >= 0 = member(x',xs) member[Ite][True][Ite](True(),x,xs) = 0 >= 0 = True() notEmpty(Cons(x,xs)) = 0 >= 0 = True() notEmpty(Nil()) = 0 >= 0 = False() * Step 3: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: member(x,Nil()) -> False() member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x',x),x',Cons(x,xs)) notEmpty(Cons(x,xs)) -> True() notEmpty(Nil()) -> False() - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) goal(x,xs) -> member(x,xs) member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs) member[Ite][True][Ite](True(),x,xs) -> True() - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ,goal,member,member[Ite][True][Ite] ,notEmpty} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): The following argument positions are considered usable: uargs(member[Ite][True][Ite]) = {1} Following symbols are considered usable: {!EQ,goal,member,member[Ite][True][Ite],notEmpty} TcT has computed the following interpretation: p(!EQ) = 0 p(0) = 0 p(Cons) = 0 p(False) = 0 p(Nil) = 1 p(S) = 1 p(True) = 0 p(goal) = 0 p(member) = 0 p(member[Ite][True][Ite]) = x1 p(notEmpty) = 8 Following rules are strictly oriented: notEmpty(Cons(x,xs)) = 8 > 0 = True() notEmpty(Nil()) = 8 > 0 = False() Following rules are (at-least) weakly oriented: !EQ(0(),0()) = 0 >= 0 = True() !EQ(0(),S(y)) = 0 >= 0 = False() !EQ(S(x),0()) = 0 >= 0 = False() !EQ(S(x),S(y)) = 0 >= 0 = !EQ(x,y) goal(x,xs) = 0 >= 0 = member(x,xs) member(x,Nil()) = 0 >= 0 = False() member(x',Cons(x,xs)) = 0 >= 0 = member[Ite][True][Ite](!EQ(x',x),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) = 0 >= 0 = member(x',xs) member[Ite][True][Ite](True(),x,xs) = 0 >= 0 = True() * Step 4: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: member(x,Nil()) -> False() member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x',x),x',Cons(x,xs)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) goal(x,xs) -> member(x,xs) member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs) member[Ite][True][Ite](True(),x,xs) -> True() notEmpty(Cons(x,xs)) -> True() notEmpty(Nil()) -> False() - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ,goal,member,member[Ite][True][Ite] ,notEmpty} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): The following argument positions are considered usable: uargs(member[Ite][True][Ite]) = {1} Following symbols are considered usable: {!EQ,goal,member,member[Ite][True][Ite],notEmpty} TcT has computed the following interpretation: p(!EQ) = 0 p(0) = 2 p(Cons) = 0 p(False) = 0 p(Nil) = 2 p(S) = 1 p(True) = 0 p(goal) = 12 p(member) = 12 p(member[Ite][True][Ite]) = 12 + 8*x1 p(notEmpty) = 13 Following rules are strictly oriented: member(x,Nil()) = 12 > 0 = False() Following rules are (at-least) weakly oriented: !EQ(0(),0()) = 0 >= 0 = True() !EQ(0(),S(y)) = 0 >= 0 = False() !EQ(S(x),0()) = 0 >= 0 = False() !EQ(S(x),S(y)) = 0 >= 0 = !EQ(x,y) goal(x,xs) = 12 >= 12 = member(x,xs) member(x',Cons(x,xs)) = 12 >= 12 = member[Ite][True][Ite](!EQ(x',x),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) = 12 >= 12 = member(x',xs) member[Ite][True][Ite](True(),x,xs) = 12 >= 0 = True() notEmpty(Cons(x,xs)) = 13 >= 0 = True() notEmpty(Nil()) = 13 >= 0 = False() * Step 5: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x',x),x',Cons(x,xs)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) goal(x,xs) -> member(x,xs) member(x,Nil()) -> False() member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs) member[Ite][True][Ite](True(),x,xs) -> True() notEmpty(Cons(x,xs)) -> True() notEmpty(Nil()) -> False() - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ,goal,member,member[Ite][True][Ite] ,notEmpty} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): The following argument positions are considered usable: uargs(member[Ite][True][Ite]) = {1} Following symbols are considered usable: {!EQ,goal,member,member[Ite][True][Ite],notEmpty} TcT has computed the following interpretation: p(!EQ) = 1 p(0) = 4 p(Cons) = 1 + x1 + x2 p(False) = 1 p(Nil) = 0 p(S) = x1 p(True) = 0 p(goal) = 14 + 12*x2 p(member) = 13 + 8*x2 p(member[Ite][True][Ite]) = 8*x1 + 8*x3 p(notEmpty) = 1 + 8*x1 Following rules are strictly oriented: member(x',Cons(x,xs)) = 21 + 8*x + 8*xs > 16 + 8*x + 8*xs = member[Ite][True][Ite](!EQ(x',x),x',Cons(x,xs)) Following rules are (at-least) weakly oriented: !EQ(0(),0()) = 1 >= 0 = True() !EQ(0(),S(y)) = 1 >= 1 = False() !EQ(S(x),0()) = 1 >= 1 = False() !EQ(S(x),S(y)) = 1 >= 1 = !EQ(x,y) goal(x,xs) = 14 + 12*xs >= 13 + 8*xs = member(x,xs) member(x,Nil()) = 13 >= 1 = False() member[Ite][True][Ite](False(),x',Cons(x,xs)) = 16 + 8*x + 8*xs >= 13 + 8*xs = member(x',xs) member[Ite][True][Ite](True(),x,xs) = 8*xs >= 0 = True() notEmpty(Cons(x,xs)) = 9 + 8*x + 8*xs >= 0 = True() notEmpty(Nil()) = 1 >= 1 = False() * Step 6: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) goal(x,xs) -> member(x,xs) member(x,Nil()) -> False() member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x',x),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs) member[Ite][True][Ite](True(),x,xs) -> True() notEmpty(Cons(x,xs)) -> True() notEmpty(Nil()) -> False() - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ,goal,member,member[Ite][True][Ite] ,notEmpty} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))